Maxwell's equations are a system of partial differential equations in 3D that connect the electric and magnetic fields observed either in the vacuum or in actual materials (such as seawater and subsurface rocks) resulting from applied electric or magnetic charges and currents. For completeness, it may be noted that the fundamental magnetic charge (the so-called magnetic monopole) does not actually exist in nature, unlike the fundamental electric charge, however, Maxwell's equations provide a mathematical description of the effects magnetic monopole charges would generate if they existed. The sources or generators of electric or magnetic currents are called transmitters. Electric and magnetic fields are vector quantities in the sense that they possess both magnitude (length) and direction in 3D. They are typically described mathematically in terms of their components along each of the three orthogonal directions of an agreed upon Cartesian coordinate system for which all directions are oriented with respect to the other two directions according to a right-hand rule. Such systems are referred to as right-hand systems. In general, the applied transmitter currents vary with time in which case the electric and magnetic fields are coupled to each other and will also vary with time even if the properties of the medium do not vary. The special cases of static applied electric or magnetic charge distributions in a static medium are referred to as electrostatics and magnetostatics. In the static cases, electric and magnetic fields are not coupled to each other mathematically by Maxwell's equations. When the applied time varying electric and magnetic currents are rich in only certain frequency components, Maxwell's equations may be usefully transformed to the frequency domain by application of the Fourier transform. In the frequency domain, the electric and magnetic fields become complex quantities which depend upon the frequency as well as upon the locations and directions of the applied electric and magnetic currents and upon the location of the point (in 3D) at which the fields are measured.
As applied to hydrocarbon exploration, electromagnetic measurements made from the surface or from the near ocean bottom with time varying applied electric current sources can be used, under favorable conditions, to reveal information about subsurface resistive structures. The property or ability of a unit volume of material to conduct electricity is measured in Siemens per meter (S/m). The ability of a particular sample to conduct a flow of electric current along a particular direction is proportional to the surface area provided by the sample to the electric current flow and is inversely proportional to the length of the sample along the current flow direction. Resistivity is the reciprocal of conductivity and is measured in ohm-meters. Thus the previous statement can be recognized as the familiar rule of circuits for resistors wired in parallel. Resistivity information can be of great value in prospecting for hydrocarbons in many important locations because oil and/or gas saturated reservoirs often exhibit strong contrasts in resistivity compared to surrounding brine filled rock structures. This is particularly true of clastic rock materials which are overwhelmingly composed of sandstones, silts and shale. In such situations, sandstone units, which are relatively porous, provide the reservoir materials. Reservoir fluid saturations comprise all possible mixtures of brine, fresh water, gas and oil; all displaying relatively predictable resistivity properties (based on the volume fraction) with highly oil saturated reservoir materials exhibiting resistivities as high as 1000 ohm-m. In contrast, brine filled rocks have resistivities in the range of 1 ohm-m. In addition, the deepwater marine situation is particularly favorable for CSEM technology because of the follow reasons:
1. Deepwater exploration wells are extremely expensive (10 to 30 million USD), making additional subsurface information concerning the presence (or absence) of commercially exploitable hydrocarbons of great commercial value.
2. Seawater, which is highly conductive (˜5 S/m), provides excellent coupling to the transmitting generators and receiving detectors.
3. The sea bottom in deepwater situations (determined by the frequency of the applied current), provides excellent screening of electromagnetic signals which would otherwise act as sources of noise.
4. The air to water interface, being an interface between the non-conductive air (conductivity ˜10−10 S/m) and relatively conductive seawater, acts to reflect electromagnetic energy radiated upward by the transmitting generator back downward towards the targets of interest, effectively doubling the transmitter efficiency.
To be of practical value in hydrocarbon exploration relatively inexpensive measurements made at the surface or ocean bottom must usefully constrain subsurface resistivity features in a manner that assists in reducing exploration risk. This is an inverse problem in which the electromagnetic data are used to determine valuable aspects of the subsurface resistivities in a process typically called data inversion. In general one or both of the following two procedures are used.
1. Human guided, by-hand adjustment of the 3D resistivity model is used until a sufficiently good match is obtained between relevant aspects of measured and synthesized electromagnetic data. This process usually begins with a relatively sophisticated subsurface model based upon an interpretation made with conventional seismic data together with an hypothesis for converting seismic attributes, such as amplitude, to predictions of resistivity. Obviously this procedure benefits from actual well measurements of resistivity which can be usefully extrapolated into the area of interest. This approach also relies upon the ability to simulate solutions of Maxwell's equations in realistic situations. The term simulate in this context—sometimes called forward modeling—means to solve Maxwell's equations by numerical methods for one or more electromagnetic field components, using assumed resistivity values as a function of position throughout the region of interest (the resistivity model). (Analytical solutions exist for Maxwell's equations only for the simplest resistivity models.) Realistic situations will typically present 3D geometry, likely including variations in water bottom topography and resistivity. Since MCSEM data are typically collected with generating transmitters located about 50 m above the water bottom, and with detectors located on the water bottom, sea bottom changes in resistivity can have significant effects upon the actual measured electromagnetic fields.
2. Inversion procedures that can be fully automated in which an appropriate mathematical measure of the misfit difference between measured and processed actual CSEM data is reduced by a numerical optimization procedure that adjusts subsurface resistivities (or equivalently conductivities) and possibly other important parameters (such as applied current magnitudes or phase) until the misfit difference is reduced to a sufficiently small value (relative to the expected noise level in the measured and processed data and relative to the expected noise level expected in the simulated data as well). It should be noted that all actual 3D CSEM datasets are insufficient to uniquely determine subsurface resistivities in a strict mathematical sense even on reasonable distance scales in portions of the subsurface that are well illuminated by the transmitting generators. In all cases, inverted resistivity models represent a few of the many possible realizations of actual resistivity distributions in the real earth. These inverted models produce simulations that resemble to a greater or lesser extent the measured and processed data. Thus, non-uniqueness and the need for expert interpretation remain significant issues. However, inversion studies of CSEM data can clearly reduce real world hydrocarbon exploration risks, particularly when combined with other forms of geophysical data. Inversion procedures applied to Maxwell's equations require cycles of simulation and back propagation amounting to hundreds of re-simulations of the measured and processed data. Back propagation refers to a mathematical procedure which produces a computationally efficient means of computing the derivatives of the inversion misfit error with respect to the model resistivities or conductivities. Derivative information of this kind is used by the automatic optimization process to reduce the misfit error.
Both of the methodologies outlined above for inverting MCSEM survey data to estimate subsurface resistivity structure for hydrocarbon exploration require repeated solution of Maxwell's equations in 3D. This is a formidable computational undertaking requiring the best possible computer implementation. As will be explained below, the present invention arranges the solution and back projection computations in such a manner that processing units can be organized in a manner that efficiently allows extremely large number of processing units to be effectively deployed. Previously reported solution schemes (see, for example, Newman and Alumbaugh, “Three-dimensional massively parallel electromagnetic inversion 1,” Geophysics Journal International 128, 345-354 (1997)) allow the use of multiple processing units by assigning regions of the modeling domain to the processing units based upon mapping the 3D resistivity or conductivity model on to the available processors organized into a 3D mesh. Thus, a 3D resistivity or conductivity model composed of 200 cells along the X, Y, Z Cartesian directions (a volume of 2003 cells) might be mapped onto 4×4×4=64 processing units by assigning 200/4=50 cells to each task along each of X,Y,Z. Solution of Maxwell's equations will in general involve solution of an algebraic system of equations obtained by the application of an appropriate discretization rule to the underlying system of partial differential equations. When using a parallel implementation, solution methods are preferably selected that lend themselves naturally to parallel implementations. Consider the case in which the magnetic fields are eliminated from Maxwell's equations in favor of the electric fields. This is acceptable provided that one is interested in the case of applied electric and magnetic currents which vary in time (the non-DC case). (It may be recalled that for DC sources which are governed by electrostatics and magnetostatics, the electric and magnetic fields are not coupled). The resulting equations involve second order partial derivatives with respect to the three Cartesian coordinates X,Y,Z. Within a second order finite difference scheme, one obtains a system of algebraic equations for the electric fields along the cell edges due to the applied electric currents,Ke=jwhere K is a square matrix of rank equal to three times the total number of cells (2003 for example). This is the number of electric field unknowns because the electric field is a vector with three components. The unknown electric fields are organized into a column vector (e in the above equation) of length three times the number of cells. The right hand side (j in the above equation), again is a column vector of length three times the number of cells and represents the electric currents created by the generating transmitters. Fortunately the matrix K has zero elements almost everywhere. The non-zero entries can be contained within 13 column vectors of length three times the number of cells. This compact representation also requires a set of index vectors of the same size to map back and forth between the original elements of K and the compactly stored non-zero values. Solutions for the electric fields can be obtained by an iterative solution method which involves repeated multiplication of K by residual vectors constructed from the difference between K applied to the solution for the current iteration and the right hand side of the above equation. These solution procedures lend themselves to parallel implementation because, when applied in a framework where the original 3D modeling domain has been assigned to, for example 43=64 separate processing tasks, each task has only 503 cells assigned to it, and multiplication of K by the residual vectors involves communication of only data along the domain boundaries (this is surface data along the faces of 502 cells in the example).